I think 5 days in Paris, if you have been there before, is fine. But if it is your first time, I think that is not enough time. I would stay in Paris the whole time and take 2 or 3 day trips. Then you have more time in Paris, but get to see other things as well. Also, you lose time traveling.
With that being said, and something I did not know on my last trip, you HAVE to go to Notre Dame. And it now requires a reservation unless you want to wait in a massive queue (I chose not to knowing I would be back in Paris soon).
If you are doing Notre Dame, you should hit up Sainte Chapelle as well. There is a great restaurant right near Notre Dame (Les Deux Colombes) where I have whiled away a few hours with a bottle of wine.
For hotels, my go to is Le Roch Hotel & Spa in the 1st, though I want to stay in Hotel Marquis Faubourg Saint-Honore (in the 8th but just...right where Rue Saint-Honore flips from the 1st to the 8th) and Voltire (also on Rue St. Roch near the first hotel).
I have a lot of other recs, but I need to sit down with them.